Problem 35 - Thinking Mathematically

Question

Construct a truth table for the given statement. \((p \wedge \sim q) \vee(p \wedge q)\)

Step-by-Step Solution

Answer

Truth table: p = [T, T, F, F], q = [T, F, T, F], \(\sim q = [F, T, F, T], p \wedge \sim q = [F, T, F, F], p \wedge q = [T, F, F, F], (p \wedge \sim q) \vee(p \wedge q) = [T, T, F, F].

Step 1: List all possible values for p and q

As there are two variables, there would be \(2^2 = 4\) possible combinations of true (T) and false (F) values. List these in a table.

Step 2: Calculate \(\sim q\)

The value of \(\sim q\) will be true if q is false, and vice versa. Add a column for \(\sim q\) in the truth table.

Step 3: Calculate \(p \wedge \sim q\) and \(p \wedge q\)

Using the values of p, q, and \(\sim q\), calculate the truth values of \(p \wedge \sim q\) and \(p \wedge q\). The AND operation is true if and only if both operands are true.

Step 4: Calculate \((p \wedge \sim q) \vee(p \wedge q)\)

Using the calculated values of \(p \wedge \sim q\) and \(p \wedge q\), calculate the truth value for the entire statement \((p \wedge \sim q) \vee(p \wedge q)\). The OR operation is true if either operand is true. This will be the final column on the truth table.

Key Concepts

Logical Conjunction

In logic, the term logical conjunction refers to the AND operation, typically represented by the symbol \ or sometimes by a simple multiplication symbol. It relates to a function between two variables or propositions that results in a truth value of true only when both propositions are true. A common way to visualize this is through a truth table, which displays all possible combinations of truth values for the variables involved.

For instance, if we have two propositions, p and q, their conjunction, denoted as \(p \wedge q\), would only yield a true result if both p is true and q is true. In all other cases, the result will be false.

• If p is true and q is true, then \(p \wedge q\) is true.
• If p is true and q is false, then \(p \wedge q\) is false.
• If p is false and q is true, then \(p \wedge q\) is false.
• If p is false and q is false, then \(p \wedge q\) is false.

Understanding this basic logical function is crucial in studying more complex logical expressions in mathematics and computer science.

Logical Disjunction

Contrasting with conjunction, the logical disjunction is another fundamental operation in logic, symbolized by \. This operation is also known as the OR operation. Logical disjunction between two statements is true if at least one of the propositions is true. It only results in a false value when both propositions are false.

For two propositions, p and q, we denote their disjunction as \(p \vee q\). The resulting truth values are determined as follows:

• If either p or q is true (or both), then \(p \vee q\) is true.
• Only if both p and q are false is \(p \vee q\) false.

In a truth table, which lists all possible combinations of truth values for p and q, the column for \(p \vee q\) will show how the logical disjunction reflects the presence of truth in at least one of the variables.

Negation in Logic

The concept of negation in logic, denoted by a tilde \(\sim\) or a NOT operator, represents a fundamental logical operation that inverts the truth value of a proposition. If a statement p is true, then its negation, written as \(\sim p\) or NOT p, is false, and vice versa.

In practice, constructing a truth table for a negation is quite simple. You only need to consider the truth value of the original proposition and then flip it:

• If p is true, then \(\sim p\) is false.
• If p is false, then \(\sim p\) is true.

The negation is a crucial operation because it allows the expression of contradiction and plays a significant role in the construction of complex logical arguments and proofs.

Logic in Mathematics

The use of logic in mathematics underpins the very foundation of mathematical reasoning and proof. Logic provides the structure and rules needed to navigate and determine the truth of mathematical statements. As a branch of philosophy, logic also interacts with mathematics in areas like set theory and algebra.

Logical operations like conjunction, disjunction, and negation form the basis of boolean algebra, which is used in digital circuit design and computer programming. Mathematics, particularly through the use of truth tables, helps us understand how we can combine different logical operations to express complex ideas in clear, unambiguous ways. This understanding is crucial not only in academics but also in real-world applications, such as developing algorithms and performing logical analyses.